Mr. small's eighth grade math class -...

Common Core Learning StandardsGRADE 8 Mathematics

GEOMETRY

Common Core Learning Standards Concepts Embedded Skills Vocabulary

Understand congruence and similarity using physical models, transparencies, or geometry software.

Transformations Translate, rotate, and reflect lines and line segments.

Rotation Reflection Translation Congruence Transformation Corresponding

parts

Explain the preservation of the sides of a figure through a given transformation.

8.G.1. Verify experimentally the properties of rotations, reflections, and translations:8.G.1a. Lines are taken to lines, and line segments to line segments of the same length.

Identify corresponding parts between a figure and its image using prime notation.Show that lines are taken to lines and line segments are taken to line segments.

SAMPLE TASKSI.) Describe the given notation with its appropriate transformation.

a) D2 _________________________________________________________________________________

b) (x−6 , y+2) _________________________________________________________________________________

c) r y= x __________________________________________________________________________________

d) R90o ___________________________________________________________________________________

Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

I.) On the given coordinate plane, graph and label the line segment whose endpoints are A(1, 4) and B(5, 3).Translate AB 3 units right and 4 units down, label your image A' B' .

Name and explain what property is preserved under the translation.

_____________________________________________________

_____________________________________________________

_______________________________________________________

_______________________________________________________

II.)


What type of transformation is being demonstrated?

_________________________________

Under this transformation, H 'T ' is the image of HT .

Is orientation of the line segment preserved? Explain your answer.

_________________________________________________________________

Is the size of the line segment preserved? Explain your answer.



Transformations Translate, rotate, and reflect geometric shapes on a coordinate plane.

Rotation Reflection Translation Congruence Properties Transformation Corresponding

parts

Measure angles using a protractor.

8.G.1. Verify experimentally the properties of rotations, reflections, and translations:8.G.1b. Angles are taken to angles of the same measure.

Identify corresponding parts between a figure and its image using prime notation.Show that angles are taken to angles of the same measure.

SAMPLE TASKS


I.)

II.)


In the given transformation, identify the congruent angle pairs from original quadrilateral MATH to its image M ' A' T ' H '.

Angle ____________ ≅ Angle _______________

Angle ____________ ≅ Angle _______________

Angle ____________ ≅ Angle _______________

Angle ____________ ≅ Angle _______________

What mathematical instrument could you use to explain why your angle pairs are congruent?

_________________________________________________________

_________________________________________________________

CB

A

Given the clockwise rotation, label the appropriate angles with A’, B’, and C’.

Explain why you chose the labels for each angle.

____________________________________________

____________________________________________

____________________________________________



Transformations Translate, rotate, and reflect parallel lines on a coordinate plane.

Parallel lines Rotation Reflection Translation Transformation Slope/rate of

change

Explain that the slope and distance between two parallel lines will be preserved after a translation, rotation, or reflection of the parallel lines.

8.G.1. Verify experimentally the properties of

Identify corresponding lines after a translation, rotation, or reflection, by using prime notation.


rotations, reflections, and translations:8.G.1c. Parallel lines are taken to parallel lines.

Show that parallel lines are translated, rotated, or reflected parallel lines.

SAMPLE TASKSI.

II.) If two parallel lines are reflected over the y-axis, what characteristics of the original parallel lines are preserved after the reflection? Sketch a diagram to help support your answer.


Translate lines a and b, with the motion rule (x +2, y – 1). Label the appropriate images lines a’ and b’.

Find the slopes of your translated images.

slope of line a’ _________ slope of line b’ ___________

How do the slopes of your transformations compare with the slopes of the given parallel lines?

_________________________________________________________

_________________________________________________________

line a line b

Characteristics Preserved:

________________________________________________

________________________________________________

________________________________________________



Congruent figures

Explain the preservation of congruence when a figure is rotated, reflected, and/or translated.

Transformation Reflection Rotation Translation Congruence Corresponding

parts

Describe the sequence of transformations that occurred from the original 2D figure to the image.

8.G.2. Understand that a two-dimensional figure is congruent to another if the second can be

Draw a reflection of an object.Draw a translation of an object.Draw a rotation of an object.


obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Name corresponding parts in congruent figures.

SAMPLE TASKS Figure 1 is transformed using two steps to form figure 2, as shown on this coordinate plane.

Part A: Complete the statements below to show which corresponding sides in each figure are congruent.

Part B: Describe the two steps used to complete this transformation.

Step 1: _________________________________________________________________

Step 2: _________________________________________________________________

Part AUsing the coordinate grid below, graph the triangle that has vertices of A(-8,4), B(-6,6), and C(-4,2).

Part BRotate ∆ABC 180° and label the image on the same coordinate grid.

Part CDilate the image from Part A by a scale factor of ½ and label the new image appropriately.




Transformations Dilate a two-dimensional figure using coordinates. Coordinate Figure Ordered pair Reflect Translate Dilate

Describe the effect of dilating a two-dimensional figure using coordinates.

8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional

Rotate a two-dimensional figure using coordinates.Describe the effect of rotating a two-dimensional figure using coordinates.


figures using coordinates. Translate a two-dimensional figure using coordinates.Describe the effect of translating a two-dimensional figure using coordinates.Reflect a two-dimensional figure using coordinates.Describe the effect of reflecting a two-dimensional figure using coordinates.

SAMPLE TASKSSee Below

I. Dilate the figure below with a scale factor of 12 . Label the image using A’, B’, and C’.


A

B

C

Explain how you found the coordinates A’, B’, and C’.

II. Part APlot the coordinates J(-2,8), K(-4,8), L(-7,2), M(-2,2) on the coordinate plane below.


Part BReflect trapezoid JKML over the line y = x. Label the image coordinates J’, K’,L’, and M’.

III. The image resulting from the translation 5 units left and 2 units up has the coordinates J’(-9,6), K’(-6,6), L’(-3,2) and M’(-6,2).


Describe the effect of a reflection over the line y = x of any coordinate (x,y).

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

Part A Determine the coordinates of the original parallelogram. Graph and label J,K,M, and L on the axis above.

Part B Explain how you determined the location of coordinate K.

IV. Rotate parallelogram MATH 90o clockwise around the origin on the coordinate plane below. Label M’ A’, T’, and H’.


V. Trinity was asked to rotate figure ABC 180o clockwise. Her answer, A’B’C’ is below.

Did Trinity correctly rotate the figure? Explain your reasoning.


Figure A

Figure C

Figure B

What two transformations were used to get from figure A to figure C?

Compare figure A with figure C. Have any of the characteristics of the figure changed after the two transformations? Explain.



Similarity with transformations

Explain the preservation of similarity when a figure is dilated, rotated, reflected, and/or translated.

Rotation Reflection Translation Dilation Transformation Similarity Congruent Similar

Describe the sequence of transformations that occurred from the original 2D figure to the image to show the similarity.

8.G.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

SAMPLE TASKSI.


A B C

D E

62°58°



Prove/explain why the three angles of a triangle equal 180°.

Triangle Similar Parallel lines Transversal Congruent Supplementar

y Linear pair Corresponding Vertical Alternate,

exterior, interior angles

Prove/explain why the exterior angle theorem of a triangle.

8.G.5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Prove/explain why alternate interior angles are congruent.Prove/explain why alternate exterior angles are congruent.Prove/explain why corresponding angles are congruent.Prove/explain why angle-angle criterion works to prove similarity of two triangles.

SAMPLE TASKS Given that AC is parallel to DE, prove that the angles of triangle BED add up to .


_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

Triangle XYZ and its exterior angles are shown below.

Part A: What is ? ____________

Part B: What is ? ____________

Explain how you found .

______________________________________________________________________________________________________

______________________________________________________________________________________________________

______________________________________________________________________________________________________

In the figure below, m∠1=118° and , m∠ 6=72°.


1


s

r

Part AName a pair of vertical angles. ____________________________

Part BAre lines r and s parallel? Explain your answer.

___________________________________________________________

___________________________________________________________

___________________________________________________________

___________________________________________________________(figure NOT drawn to scale)

The measures of two angles of triangle ABC are given as 40° and 80°. The measures of two angles of triangle DEF are given as 80° and 60°.

(Triangles are NOT drawn to scale.)

Are Triangles ABC and FED similar? Explain your answer.

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________


BE

A

F80°

D

40°

60°80°

C


Understand and apply the Pythagorean Theorem.

Pythagorean theorem

Explain a proof of Pythagorean theorem. (If a triangle is a right triangle, then a2 + b2 = c2)

Pythagorean theorem

Converse Proof Legs Hypotenuse

Explain a proof of the converse of Pythagorean theorem. (If a2 + b2 = c2, then a triangle is a right triangle)

8.G.6. Explain a proof of the Pythagorean Theorem and its converse.

Identify the legs and hypotenuse of a right triangle.Solve multi-step equations.

SAMPLE TASKSI. Given triangle ABC with side lengths of 10 cm, 8 cm, and 5 cm prove whether or not the sides form a right triangle.

Show your work.

II. Three edges of cornfields meet to form a right triangle. The first field has an edge of 30 yards, the second field has an edge of 40


yards, and the third field has an edge of 50 yards. Using these dimensions, prove the Pythagorean Theorem.Show your work.

III. Jacob needs to construct a right triangle using drinking straws. If he has straw lengths of 5 cm, 12 cm and 15 cm, can he construct a right triangle? Explain why or why not.

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________


Field 1

Field 2

Field 3


Pythagorean theorem

Calculate the length of a leg of a right triangle using Pythagorean theorem.

Leg Hypotenuse Right angle Pythagorean

theorem Square root Radical Diagonals

Calculate the length of the hypotenuse of a right triangle using Pythagorean theorem.

8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Calculate the diagonal of a three-dimensional figure using Pythagorean Theorem.

Read and interpret a word problem involving Pythagorean Theorem.Solve word problems involving Pythagorean Theorem.Solve multi-step equations.Identify the legs and hypotenuse of a right triangle.

Round to given place value.

SAMPLE TASKS In Juanita’s Little League championship game the bases are all 60 feet apart forming a square. Juanita is on first base planning on

stealing second, how far does the catcher need to throw, to the nearest foot, the ball from home plate to second base in order to make the out?

Simon leans a 20-foot ladder against the side of his house so that the base of the ladder is 5 feet from the house. About how high up the side of the house does the ladder reach? Round your answer to the nearest tenth of a foot.Show your work.


60 ft.

A cylindrical can of soda has a height of 5 inches and a diameter of 2 inches as shown. There is a straw that is stuck diagonally inside the can. What is the length of this straw? Show your work.

Length of straw: ____________________________



2 in

5 in

straw


Pythagorean theorem on a

coordinate plane

Calculate the distance between two points in a coordinate plane using the Pythagorean Theorem.

Leg Hypotenuse Right angle Pythagorean

theorem Ordered pair Coordinate

plane Square root Distance

formula

8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Plot points in a coordinate plane

Solve multi-step equations

Identify the legs and hypotenuse of a right triangle

Solve Pythagorean Theorem

SAMPLE TASKS Line segment DE has endpoints at D(6, 1) and E(-6, 6). What is the length of DE? Use the grid below to help show your work.

A team of archaeologists fenced off an ancient ruin they were exploring. They created a grid to represent the area so they could label the locations of several artifacts. To the nearest tenth of a meter, how many meters apart were the pitcher (point R) and the pot


(point P)?



Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Write and solve using the formula for the volume of a cone.

Volume Cone Cylinder Sphere Area Base Formula

Write and solve using the formula for the volume of a cylinder.

8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Write and solve using the formula for the volume of a sphere.Solve word problems involving the volume of cones, cylinders, and spheres.

Solve a multi-step equation for a missing variable.

SAMPLE TASKS

Describe the relationship between the volume of a cylinder and the volume of a cone where the cylinder and the cone have the same height and diameter.

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

A pitcher holds 1,614.7 in3 of liquid. Each can of punch is 15 inches tall with a diameter of 8 inches. How many full cans will the pitcher hold?


Explain your answer.

__________________________________________________________________________________________________________

__________________________________________________________________________________________________________

__________________________________________________________________________________________________________

A cylindrical can has a height of 24 inches and a volume of 864 cubic inches. What is the length of the diameter to the nearest inch?

To the nearest cubic inch, how much space is there inside a hamster ball with a diameter of 10 inches?

Two companies manufacture tanks that store water. Company A sells cylindrical tanks like the one shown below for $500 and company B sells spherical tanks like the one shown below also for $500. Which company should you purchase your tank from in order


to maximize your water storage?Show your work.

Company ________

Explain your answer.

___________________________________________________________________________________________________________

___________________________________________________________________________________________________________

____________________________________________________________________________________________________________


8 ft

3 ft

r = 2.5 ft.

Mr. small's eighth grade math class -...

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